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Optical phenomena in photoelastic models by the rotation of principal axes

Investigation shows that, using the matrix representation of the solution of the equations of photoelasticity, an adequate description of the complicated optical phenomena in three-dimensional photoelastic models can be obtained

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Abstract

The basic equations of three-dimensional photoelasticity are derived in a form which is simpler than that of equations known previously. Using the matrix representation of the solution of these equations, it is also shown that when rotation of principal axes is present there always exist two perpendicular directions of polarizer by which the light emerging from the model is linearly polarized. These polarization directions of the incident and emergent light are named primary and secondary characteristic directions, respectively. The experimental determination of characteristic directions, as well as of the phase retardation, gives three equations on every light path to determine the stress components in a three-dimensional model. A general algorithm of the method of characteristic directions is presented, and its application by determination of stress in shells by normal and tangential incidence is described. A further extension of the method to the general axisymmetric problem has been suggested.

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Abbreviations

A 1,A 2 :

components of electric vector of light after transformation [eq (4)]

a 1,…,a k,b 1,…,b e,c 1,…,c m,d 1,…,d k :

constants which determine the distribution of stress in an axisymmetric model

B 1,B 2 :

components of electric vector of light after transformation [eq 7]

B 10,B 20 :

components of the incident-light vector on arbitrary coordinate axes

B 10 o,B 20 o :

components of the incident-light vector in primary characteristic directions

B 1 o,B 2 o :

components of the emergent-light vector in secondary characteristic directions

c :

velocity of light in vacuum

C :

\(\frac{1}{{2k}}\frac{{\omega ^2 }}{{c^2 }}\)

C 0,C 1 :

photoelastic constants

C′:

CC 0

D 1,D 2 :

components of electric-induction vector of light

E 1,E 2 :

components of electric vector of light

f 1,…,f 4 :

functions which determine the distribution of stress in an axisymmetric model

G(γ):

diagnonal unitary matrix [eq (20)]

k :

ωN/c

N :

index of refraction of the non-stressed medium

R :

\(\frac{{2\varphi _0 }}{\Delta }\)

r :

outer radius of a cylindrical shell; radial coordinate in an axisymmetric model

S :

\(\sqrt {1 + R^2 } \)

S(αj):

matrix of rotation [eq (19)]

t :

thickness of the model

U :

unitary matrix [eq (17)]

U 1 :

unitary matrix which transforms the incident-light vector into the plane of symmetry

U 1 * :

transpose ofU 1

u′, v′ :

primary characteristic directions

u″, v″ :

secondary characteristic directions

x 1 ′,x 2 ′ :

principal directions at the point of entrance of light

x 1 ″,x 2 ″ :

principal directions at the point of emergence of light

x 1,x 2,z :

rectangular coordinates

z 1 :

z 2/rt

α:

angle between conjugate characteristic directions

α1, α2 :

angles which determine the primary and secondary characteristic directions

β:

\(\frac{{3(1 - \mu ^2 )}}{{r^2 t^2 }}\)

2γ:

characteristic phase retardation which corresponds to the matrixU

2γ1 :

characteristic phase retardation which corresponds to the matrixU 1

Δ:

C't(σ1 − σ2)

Δ*:

phase retardation determined by [eq (37)]

σ ij :

Kronecker's tensor

∈ ij :

tensor of dielectric constant

∈ j :

principal components of the tensor of dielectric constant perpendicular to the wave normal

μ:

Poisson's ratio

ξ, ζ, θ:

parameters of the matrixU

σ ij :

stress tensor

σ1, σ2 :

principal stresses perpendicular to the wave normal

(σ11 − σ22) n , (σ12) n :

membrane stress components in a shell

(σ11 − σ22) b , (σ12) b :

maximum bending stresses in a shell

σ n :

½C′(σ11 − σ22) n

σ b :

½C′(σ11 − σ22) b

σ2b :

longitudinal bending stress in a cylindrical shell

σϑb :

circumferential bending stress in a cylindrical shell

σϑn :

circumferential membrane stress in a cylindrical shell

σ2n :

longitudinal membrane stress in a cylindrical shell

σ2b °:

max σ2b

σ:

\({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}C'\left( {\frac{t}{r}\sigma \vartheta _n - \sigma _{2b} ^\circ } \right)\)

σ r , σϑ, σ22, τ2r :

stress components in an axisymmetric model

τ n :

C′(σ12) n

τ2r :

shearing stress in a cylindrical shell

τ1r °:

max τ1r

τ:

C′τ2r °

ϕ:

angle of rotation of principal axes

ϑ0 :

total angle of rotation of principal axes

ψ:

1/2SΔ

ω:

frequency of vibration of light

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  1. Department of Applied Mathemtics and Mechanics, Institute of Cybernetics, Academy of Sciences of the Estonian SSR, Tallinn, U.S.S.R., Estonia

    Hillar K. Aben (Head)

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Aben, H.K. Optical phenomena in photoelastic models by the rotation of principal axes. Experimental Mechanics 6, 13–22 (1966). https://doi.org/10.1007/BF02327109

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